American Institute of Mathematical Sciences

Advanced
May  2017, 37(5): 2881-2897. doi: 10.3934/dcds.2017124

Persistence and stationary distribution of a stochastic predator-prey model under regime switching

Li Zu 1, , Daqing Jiang 2,3,, and  Donal O'Regan 4,

1. 

College of Mathematics and Statistics, Hainan Normal University, Hainan 571158, China
2. 

School of Science, China University of Petroleum (East China), Qingdao 266580, China
3. 

Nonlinear Analysis and Applied Mathematics(NAAM)-Research Group, King Abdulaziz University, Jeddah, Saudi Arabia
4. 

School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

* Corresponding author: Daqing Jiang (School of Science, China University of Petroleum (East China), Qingdao), China E-mail: daqingjiang2010@hotmail.com, Tel.: +86 43185099589; fax: +86 43185098237

Received  October 2014 Revised  January 2017 Published  February 2017

Fund Project: The first author is supported by NSFC of China (No: 11601038), the second author is supported by NSFC of China (No: 11371085) and the Fundamental Research Funds for the Central Universities (No: 15CX08011A).

Taking both white noise and colored environment noise into account, a predator-prey model is proposed. In this paper, our main aim is to study the stationary distribution of the solution and obtain the threshold between persistence in mean and the extinction of the stochastic system with regime switching. Some simulation figures are presented to support the analytical findings.

Keywords: Stochastic predator-prey model, threshold, Markov switching, stationary distribution, extinction.
Mathematics Subject Classification: Primary: 34F05, 92D25; Secondary: 92Bxx.
Citation: Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124
References:
[1]

G. Q. Cai and Y. K. Lin, Stochastic analysis of predator-prey type ecosystems, Ecol. Complex., 4 (2007), 242-249. doi: 10.1016/j.ecocom.2007.06.011. Google Scholar

[2]

L. S. Chen and Z. J. Jing, The existence and uniqueness of limit cycles for the differential equations of predator-prey interactions, Chinese Sci. Bull., 9 (1984), 521-523. Google Scholar

[3]

H. W. Hethcote, W. Wang, L. T. Han and Z. E. Ma, A predator-prey model with infected prey, Theor. Popul. Biol., 66 (2004), 259-268. doi: 10.1016/j.tpb.2004.06.010. Google Scholar

[4]

C. Y. Ji, D. Q. Jiang and N. Z. Shi, A note on a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 377 (2011), 435-440. doi: 10.1016/j.jmaa.2010.11.008. Google Scholar

[5]

R. Z. Khasminskii, C. Zhu and G. Yin, Stability of regime-switching diffusions, Stochastic Process. Appl., 117 (2007), 1037-1051. doi: 10.1016/j.spa.2006.12.001. Google Scholar

[6]

H. H. Li, D. Q. Jiang, F. Z. Cong and H. X. Li, Persistence and non-persistence of a predator-prey system with stochastic perturbation Abstr. Appl. Anal. 2014 (2014), Article ID 720283, 10 pages. doi: 10.1155/2014/720283. Google Scholar

[7]

X. Y. Li, D. Q. Jiang and X. R. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448. doi: 10.1016/j.cam.2009.06.021. Google Scholar

[8]

X. Y. Li, A. Gray, D. Q. Jiang and X. R. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11-28. doi: 10.1016/j.jmaa.2010.10.053. Google Scholar

[9]

X. Y. Li and X. R. Mao, A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching, Automatica, 48 (2012), 2329-2334. doi: 10.1016/j.automatica.2012.06.045. Google Scholar

[10]

H. Liu, Q. S. Yang and D. Q. Jiang, The asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidences, Automatica, 48 (2012), 820-825. doi: 10.1016/j.automatica.2012.02.010. Google Scholar

[11]

H. Liu, X. X. Li and Q. S. Yang, The ergodic property and positive recurrence of a multi-group Lotka-Volterra mutualistic system with regime switching, Syst. Control Lett., 62 (2013), 805-810. doi: 10.1016/j.sysconle.2013.06.002. Google Scholar

[12]

M. Liu and K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching Ⅱ, Math. Comput. Model., 55 (2012), 405-418. doi: 10.1016/j.mcm.2011.08.019. Google Scholar

[13]

M. Liu and K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching, Math. Comput. Model., 54 (2011), 2139-2154. doi: 10.1016/j.mcm.2011.05.023. Google Scholar

[14]

X. R. Mao, S. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156. doi: 10.1016/S0022-247X(03)00539-0. Google Scholar

[15]

X. R. Mao, Stationary distribution of stochastic population systems, Syst. Control Lett., 60 (2011), 398-405. doi: 10.1016/j.sysconle.2011.02.013. Google Scholar

[16]

X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markovian Switching Imperial College Press, London, 2006. Google Scholar

[17]

R. M. Nisbet and W. S. C. Gurney, Modelling Fluctuating Populations Wiley-Interscience, New York, 1982.Google Scholar

[18]

E. C. Pielou, Introduction to Mathematical Ecology Wiley-Interscience, New York, 1969.Google Scholar

[19]

A. Settati and A. Lahrouz, Stationary distribution of stochastic population systems under regime switching, Appl. Math. Comput., 244 (2014), 235-243. doi: 10.1016/j.amc.2014.07.012. Google Scholar

[20]

Y. Takeuchi, N. H. Dub, N. T. Hieu and K. Satoa, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957. doi: 10.1016/j.jmaa.2005.11.009. Google Scholar

[21]

D. Y. Xu, B. Li, S. J. Long and L. Y. Teng, Moment estimate and existence for solutions of stochastic functional differential equations, Nonlinear Anal., 108 (2014), 128-143. doi: 10.1016/j.na.2014.05.004. Google Scholar

[22]

D. Y. Xu, X. H. Wang and Z. G. Yang, Further results on existence-uniqueness for stochastic functional differential equations, Sci. China Math., 56 (2013), 1169-1180. doi: 10.1007/s11425-012-4553-1. Google Scholar

[23]

T. Zhao, Y. Kung and H. L. Smith, Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems, Nonlinear Anal. Theor., 28 (1997), 1373-1394. doi: 10.1016/0362-546X(95)00230-S. Google Scholar

[24]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, Siam. J. Control Optim., 46 (2007), 1155-1179. doi: 10.1137/060649343. Google Scholar

[25]

L. Zu, D. Q. Jiang and D. O'Regan, Conditions for persistence and ergodicity of a stochastic Lotka-Volterra predator-prey model with regimes witching, Commun. Nonlinear Sci. Numer. Simul., 29 (2015), 1-11. doi: 10.1016/j.cnsns.2015.04.008. Google Scholar

show all references

References:
[1]

G. Q. Cai and Y. K. Lin, Stochastic analysis of predator-prey type ecosystems, Ecol. Complex., 4 (2007), 242-249. doi: 10.1016/j.ecocom.2007.06.011. Google Scholar

[2]

L. S. Chen and Z. J. Jing, The existence and uniqueness of limit cycles for the differential equations of predator-prey interactions, Chinese Sci. Bull., 9 (1984), 521-523. Google Scholar

[3]

H. W. Hethcote, W. Wang, L. T. Han and Z. E. Ma, A predator-prey model with infected prey, Theor. Popul. Biol., 66 (2004), 259-268. doi: 10.1016/j.tpb.2004.06.010. Google Scholar

[4]

C. Y. Ji, D. Q. Jiang and N. Z. Shi, A note on a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation, J. Math. Anal. Appl., 377 (2011), 435-440. doi: 10.1016/j.jmaa.2010.11.008. Google Scholar

[5]

R. Z. Khasminskii, C. Zhu and G. Yin, Stability of regime-switching diffusions, Stochastic Process. Appl., 117 (2007), 1037-1051. doi: 10.1016/j.spa.2006.12.001. Google Scholar

[6]

H. H. Li, D. Q. Jiang, F. Z. Cong and H. X. Li, Persistence and non-persistence of a predator-prey system with stochastic perturbation Abstr. Appl. Anal. 2014 (2014), Article ID 720283, 10 pages. doi: 10.1155/2014/720283. Google Scholar

[7]

X. Y. Li, D. Q. Jiang and X. R. Mao, Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232 (2009), 427-448. doi: 10.1016/j.cam.2009.06.021. Google Scholar

[8]

X. Y. Li, A. Gray, D. Q. Jiang and X. R. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl., 376 (2011), 11-28. doi: 10.1016/j.jmaa.2010.10.053. Google Scholar

[9]

X. Y. Li and X. R. Mao, A note on almost sure asymptotic stability of neutral stochastic delay differential equations with Markovian switching, Automatica, 48 (2012), 2329-2334. doi: 10.1016/j.automatica.2012.06.045. Google Scholar

[10]

H. Liu, Q. S. Yang and D. Q. Jiang, The asymptotic behavior of stochastically perturbed DI SIR epidemic models with saturated incidences, Automatica, 48 (2012), 820-825. doi: 10.1016/j.automatica.2012.02.010. Google Scholar

[11]

H. Liu, X. X. Li and Q. S. Yang, The ergodic property and positive recurrence of a multi-group Lotka-Volterra mutualistic system with regime switching, Syst. Control Lett., 62 (2013), 805-810. doi: 10.1016/j.sysconle.2013.06.002. Google Scholar

[12]

M. Liu and K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching Ⅱ, Math. Comput. Model., 55 (2012), 405-418. doi: 10.1016/j.mcm.2011.08.019. Google Scholar

[13]

M. Liu and K. Wang, Asymptotic properties and simulations of a stochastic logistic model under regime switching, Math. Comput. Model., 54 (2011), 2139-2154. doi: 10.1016/j.mcm.2011.05.023. Google Scholar

[14]

X. R. Mao, S. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287 (2003), 141-156. doi: 10.1016/S0022-247X(03)00539-0. Google Scholar

[15]

X. R. Mao, Stationary distribution of stochastic population systems, Syst. Control Lett., 60 (2011), 398-405. doi: 10.1016/j.sysconle.2011.02.013. Google Scholar

[16]

X. R. Mao and C. G. Yuan, Stochastic Differential Equations with Markovian Switching Imperial College Press, London, 2006. Google Scholar

[17]

R. M. Nisbet and W. S. C. Gurney, Modelling Fluctuating Populations Wiley-Interscience, New York, 1982.Google Scholar

[18]

E. C. Pielou, Introduction to Mathematical Ecology Wiley-Interscience, New York, 1969.Google Scholar

[19]

A. Settati and A. Lahrouz, Stationary distribution of stochastic population systems under regime switching, Appl. Math. Comput., 244 (2014), 235-243. doi: 10.1016/j.amc.2014.07.012. Google Scholar

[20]

Y. Takeuchi, N. H. Dub, N. T. Hieu and K. Satoa, Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323 (2006), 938-957. doi: 10.1016/j.jmaa.2005.11.009. Google Scholar

[21]

D. Y. Xu, B. Li, S. J. Long and L. Y. Teng, Moment estimate and existence for solutions of stochastic functional differential equations, Nonlinear Anal., 108 (2014), 128-143. doi: 10.1016/j.na.2014.05.004. Google Scholar

[22]

D. Y. Xu, X. H. Wang and Z. G. Yang, Further results on existence-uniqueness for stochastic functional differential equations, Sci. China Math., 56 (2013), 1169-1180. doi: 10.1007/s11425-012-4553-1. Google Scholar

[23]

T. Zhao, Y. Kung and H. L. Smith, Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems, Nonlinear Anal. Theor., 28 (1997), 1373-1394. doi: 10.1016/0362-546X(95)00230-S. Google Scholar

[24]

C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, Siam. J. Control Optim., 46 (2007), 1155-1179. doi: 10.1137/060649343. Google Scholar

[25]

L. Zu, D. Q. Jiang and D. O'Regan, Conditions for persistence and ergodicity of a stochastic Lotka-Volterra predator-prey model with regimes witching, Commun. Nonlinear Sci. Numer. Simul., 29 (2015), 1-11. doi: 10.1016/j.cnsns.2015.04.008. Google Scholar

Figure 1.  The curves on the subgraphs (a) and (b) are the density functions of x(t) and y(t) in k = 1, k = 2, respectively. The white noise intensity on x(t) and y(t) are all relatively small.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The subgraphs (a) and (b) denote sample phase portrait of stochastic system and the corresponding deterministic system. The red, blue and black area denote x(t), y(t) in k = 1, k = 2 and converting between the above two states, respectively. The white noise intensity on x(t) and y(t) are all relatively small.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The subgraphs (a) and (b) have the same definitions as in Fig.2. The white noise intensity on x(t) and y(t) are all relatively big.
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129
[2]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507
[3]

Nguyen Huu Du, Nguyen Thanh Dieu, Tran Dinh Tuong. Dynamic behavior of a stochastic predator-prey system under regime switching. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3483-3498. doi: 10.3934/dcdsb.2017176
[4]

Gianni Gilioli, Sara Pasquali, Fabrizio Ruggeri. Nonlinear functional response parameter estimation in a stochastic predator-prey model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 75-96. doi: 10.3934/mbe.2012.9.75
[5]

Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Amelia G. Nobile. A non-autonomous stochastic predator-prey model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 167-188. doi: 10.3934/mbe.2014.11.167
[6]

Guanqi Liu, Yuwen Wang. Stochastic spatiotemporal diffusive predator-prey systems. Communications on Pure & Applied Analysis, 2018, 17 (1) : 67-84. doi: 10.3934/cpaa.2018005
[7]

Hongxiao Hu, Liguang Xu, Kai Wang. A comparison of deterministic and stochastic predator-prey models with disease in the predator. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2837-2863. doi: 10.3934/dcdsb.2018289
[8]

Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences & Engineering, 2014, 11 (4) : 807-821. doi: 10.3934/mbe.2014.11.807
[9]

Ronald E. Mickens. Analysis of a new class of predator-prey model. Conference Publications, 2001, 2001 (Special) : 265-269. doi: 10.3934/proc.2001.2001.265
[10]

Xiaoling Li, Guangping Hu, Zhaosheng Feng, Dongliang Li. A periodic and diffusive predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 445-461. doi: 10.3934/dcdss.2017021
[11]

Yang Lu, Xia Wang, Shengqiang Liu. A non-autonomous predator-prey model with infected prey. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3817-3836. doi: 10.3934/dcdsb.2018082
[12]

Sílvia Cuadrado. Stability of equilibria of a predator-prey model of phenotype evolution. Mathematical Biosciences & Engineering, 2009, 6 (4) : 701-718. doi: 10.3934/mbe.2009.6.701
[13]

Shanshan Chen. Nonexistence of nonconstant positive steady states of a diffusive predator-prey model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 477-485. doi: 10.3934/cpaa.2018026
[14]

Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057
[15]

Haiying Jing, Zhaoyu Yang. The impact of state feedback control on a predator-prey model with functional response. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 607-614. doi: 10.3934/dcdsb.2004.4.607
[16]

Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189
[17]

Antoni Leon Dawidowicz, Anna Poskrobko. Stability problem for the age-dependent predator-prey model. Evolution Equations & Control Theory, 2018, 7 (1) : 79-93. doi: 10.3934/eect.2018005
[18]

Liang Zhang, Zhi-Cheng Wang. Spatial dynamics of a diffusive predator-prey model with stage structure. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1831-1853. doi: 10.3934/dcdsb.2015.20.1831
[19]

H. W. Broer, K. Saleh, V. Naudot, R. Roussarie. Dynamics of a predator-prey model with non-monotonic response function. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 221-251. doi: 10.3934/dcds.2007.18.221
[20]

Xiaoying Wang, Xingfu Zou. On a two-patch predator-prey model with adaptive habitancy of predators. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 677-697. doi: 10.3934/dcdsb.2016.21.677

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (12)
  • HTML views (3)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences